Bring water, lentils, and salt to a boil in a medium saucepan over high heat. Reduce the heat to medium-low and simmer, stirring occasionally, until the lentils just begin to fall apart, about 25 minutes. Drain the lentils well, transfer them to a paper towel–lined baking sheet, and gently pat them dry.

Heat oil in a 12-inch nonstick skillet over medium heat until shimmering. Add the mushrooms and cook until the released liquid has evaporated and the mushrooms are softened, about 8 minutes. Stir in the onions, celery, leek, and garlic and cook until the vegetables are softened and just beginning to brown, 10 to 15 minutes. Transfer this mixture to the sheet with the lentils and let everything cool to room temperature, about 30 minutes.

Combine water, bulgur, and salt in a large bowl and microwave, covered, until the bulgur is softened, about 5 minutes. Drain the grains well in a fine-mesh strainer, using a rubber spatula to press out every bit of excess moisture. Return the bulgur to the bowl to cool slightly. Meanwhile, pulse the cashews in a food processor until they are finely ground, about 25 pulses.

Stir the ground cashews, cooled lentil-vegetable mixture, and mayonnaise into the bowl with the bulgur until combined. Transfer half of the mixture to the food processor and pulse until coarsely ground and cohesive, about 15 pulses. Repeat with the remaining mixture, then stir in the panko, salt, and pepper until uniform.

Divide the mixture into 12 equal portions and tightly pack each into a patty about 1/2-inch thick. Heat oil in a 12-inch nonstick skillet over medium heat until shimmering. Add 4 burgers at a time and cook until well browned on the first side, about 4 minutes. Flip the burgers and cook until the second side is deeply browned, another 4 minutes. Repeat with the remaining oil and patties.

Serve warm on toasted burger buns with your preferred toppings.